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Handbooks of Mathematical Functions, Versions 1.0 and 2.0 Reviewed by Richard Beals

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Handbooks of Mathematical Functions, Versions 1.0 and 2.0 Reviewed by Richard Beals Book Review Handbooks of Mathematical Functions, Versions 1.0 and 2.0 Reviewed by Richard Beals NIST Handbook of Mathematical Functions surprise that special functions leads, with about one citation per sixty books and papers. Areas in Edited by Frank W. J. Olver, Daniel W. Lozier, the one-in-200 to one-in-900 range include integral Ronald F. Boisvert, and Charles W. Clark Cambridge University Press, 2010 transforms, number theory, most of the applied US$50.00, 966 pages, mathematics areas, almost all the science classi- ISBN: 978-05211-922-55 fications, dynamical systems, harmonic analysis, and differential and integral equations. (For mani- folds and cell complexes the rate is about one in Who in the mathematical community uses hand- 12,500.) books of special functions, and why? And will the This gives us a picture of who cites A&S. Why newest version make a difference? do they do so, apart from those who specialize By various objective measures, the handbook in the subject of special functions? Although A&S most used currently is the Handbook of Math- contains material about combinatorics, which ac- ematical Functions with Formulas, Graphs, and counts for many of the number theory citations, Mathematical Tables [AS], universally known the larger part is devoted to classical special func- as “Abramowitz and Stegun” (A&S). Estimates tions. Most of these functions arose by proposing a of the number of copies sold range up to a model physical equation and separating variables million. Since 1997, when MathSciNet began in one coordinate system or another, thereby re- compiling citation lists from papers reviewed in ducing the problem to a second-order ordinary Mathematical Reviews, A&S has been acknowl- differential equation such as Bessel’s equation. edged some 2,800 times, outdistancing other We still often find that the simplest model of compendia by a factor of two (Gradshteyn- a given physical or mathematical problem leads Ryzhik), three (Bateman Manuscript Project), five to a classical equation and, therefore, a solution (Prudnikov-Brychkov-Marichev), and more. involving special functions. In ODE this includes So, more specifically, who has been using A&S? turning points and some types of singularity. In A check of a somewhat random list of forty an- PDE it includes problems of mixed type in aero- alysts turned up a total of five such citations. dynamics and transport theory, hypoelliptic and Given the research areas and total number of degenerate elliptic problems, and weakly hyper- publications, this turns out to be close to what bolic equations. An explicit solution of a model one might predict. Per MSC major classification, problem can point the way to an understanding the largest number of citations is in numerical of more general problems, allow for calculation analysis (313 as of mid-May), followed by partial of asymptotics, and so on—provided one has at differential equations (286). Fifteen classifications hand enough information about the ingredients show one or fewer citations; manifolds and cell of the explicit solution. The list of such models complexes is one of four classifications that show continues to grow. Calculation of asymptotics also two citations. A more telling statistic is the ratio accounts for some of the usage in number theory. of papers to citations in each classification. It is no Why and how might a successor to A&S be Richard Beals is professor emeritus of mathe- useful? Over half of the thousand pages of A&S matics at Yale University. His email address is are devoted to numerical tables (which is the [email protected]. reason that this writer eyed the Dover edition September 2011 Notices of the AMS 1115 many times over many years before finally buying functions, q–hypergeometric functions, multidi- it). The ubiquity of computers and the Internet has mensional theta functions, Lamé functions, Heun made the tables largely obsolete. Most of the rest functions, 3j, 6j, 9j symbols, Painlevé transcen- consists simply of lists of identities and formulas, dents, and integrals with coalescing saddles. some graphs, and some rather general references The two methods chapters in A&S have been at the ends of chapters. There are competing expanded to three: algebraic and analytic meth- works, but not as inexpensive, convenient, and ods, asymptotic approximations, and numerical easily available: Erdélyi et al. [EMOT] runs to three methods. In addition to definitions and brief sum- volumes, and Magnus et al. [MOS], the much more maries of standard facts from real and complex comprehensive successor to [MO], has long been analysis, the first chapter introduces distributions out of print. But still, A&S is basically a cookbook. and tempered distributions and gives a number of Some users must wonder whether and how the series and integral representations of the delta dis- recipes hang together, where they come from, and tribution. The second chapter is a comprehensive how the formulas can be derived. (Such questions survey of old and new methods for asymptotics, might even drive a late arrival to the subject to with applications to differential and difference become involved in textbook writing.) There are equations. The third chapter, on numerical meth- recent textbooks and treatises, and though A&S ods, also contains new developments and is twice is cited four or more times as often as any one the size of the corresponding A&S chapter. The of them, they may well be “used” more often. So, graphics are a major advance from what was fea- again, is a successor to the point? sible at the time of A&S, mostly three-dimensional A&S was the culmination of a decade-long views in color. (See, for example, the Bessel func- project of the National Bureau of Standards, which tion graphs on pages 219–221.) Finally, the number has since been renamed the National Institute of references is about 2,300, nearly an order of of Standards and Technology (NIST). A second magnitude more than A&S, with much new work decade–long project was undertaken under NIST on asymptotics, approximations, location of zeros, auspices to update A&S. The result is a multimedia and q–analysis. successor: the print and CD-ROM NIST Handbook All this suggests that NHMF will be a very useful of Mathematical Functions (NHMF) [OLBC1] and the replacement for A&S, with a wider audience. But, online Digital Library of Mathematical Functions all in all, isn’t it basically just another cookbook? (DLMF) [OLBC2], edited by Frank W. J. Olver, The answer is yes, and no. Some cookbooks just Daniel W. Lozier, Ronald F. Boisvert, and Charles have recipes, some instruct in the art of cooking, W. Clark. and some make your mouth water. It would take a genuine expert in spe- Consider the organization of a typical chapter. cial functions—or rather a committee of such After a brief section on notation, there are sec- experts—to give an adequate appraisal of this tions on various functions, with subsections on new effort. However, many of those who are most definitions, graphics, representations and identi- knowledgeable have been directly involved in ties, zeros, integrals, sums, and asymptotics—very the new production: four editors, ten associate much in the spirit of A&S. Following this materialis editors, twenty-five validators, and twenty-nine a section on applications—mathematical and phys- authors (albeit with considerable overlap). More- ical. Sometimes brief, sometimes quite extensive, over, forty-seven members of NIST and fifty-one but generally very informative, these summaries nonmembers are acknowledged by name as having indicate areas of use, with references. A section on “contributed to the project in a variety of ways.” computation contains brief discussions of meth- Therefore it has fallen to an amateur to discuss ods, tables, and software, also with references. this production for the Notices. The final section contains general references for Except for the chapters on constants and scales the chapter, followed by specific references, listed of notation and some of the material from the subsection by subsection. For each equation, ei- chapter on probability functions, all the nontab- ther a specific reference or sketch of a derivation ular content of A&S is incorporated in NHMF, is given, sometimes noting corrections that need generally in an expanded form. This includes the to be made in the source material. elementary functions and all the usual special NHMF positively invites browsing, in the meth- functions, as well as Bernoulli and Euler polyno- ods chapters and throughout. The descriptions of mials and combinatorial analysis. The choice of applications and remarks on sources are a mine additional material has been inspired by inter- of information. Do you wonder what q-analysis is nal mathematical developments, by a broadening all about and where on earth it might be used? Do from classical analysis to related areas in alge- you want a quick view of combinatorics, Catalan bra and number theory, and by such advances in numbers, Stirling numbers, and the like? Is there mathematical physics as integrable models in con- something you use a lot, and wonder if there is tinuum mechanics and statistical physics. There anything new to be learned about it? Are you in- are new chapters on generalized hypergeometric trigued by chapters with titles like “Integrals with 1116 Notices of the AMS Volume 58, Number 8 Coalescing Saddles” or “Functions with Matrix et al. [MO], [MOS], where the principal subdivision Argument”? If you have heard of—perhaps even is by type of polynomial, the various sections here used—Bessel functions, do you have any curiosity are each arranged by topic: orthogonality rela- at all about Struwe functions and Heun functions? tions, series representations, recurrence relations, Do you know how the Chinese remainder theorem and so on.
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